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Free ISEE Math Practice Test — Grade 9

Math only, deliberately — the real ISEE (published by ERB) also has verbal and reading sections, which we don't cover.

20 real practice questions at ISEE Upper Level difficulty, drawn from the same calibrated bank our adaptive practice uses. Work through them right here — every question has the answer and a step-by-step explanation one tap away.

Question 1 · Coordinate GraphsIntro
A music streaming app tracks how a newly released song is trending. The number of new listeners per hour (in hundreds) follows a linear pattern over the days since release. On day 2, the song is gaining 5 hundred new listeners per hour. On day 6, it is losing 3 hundred listeners per hour (a value of -3 hundred). Assuming this trend is perfectly linear, how many hundred new listeners per hour was the song projected to gain at the moment of release (day 0)?
A6
B1
C9
D17
Show answer & explanation
Let's work it out
1.To find the projected value at day 0, write the equation of the line through the two known points (2, 5) and (6, -3), then evaluate it at x = 0 (which means finding the y-intercept).
2.The slope is (-3 - 5) divided by (6 - 2), which equals -8 divided by 4, or -2.
3.Substituting the point (2, 5) into y = -2x + b gives 5 = -2(2) + b, so 5 = -4 + b, and b = 9.
4.The y-intercept is 9, meaning the song was projected to be gaining 9 hundred new listeners per hour at the moment of release.
Answer: C) 9
Question 2 · Data & StatisticsIntro
At a craft club, a student makes bracelets over two sessions. In the first session, the student completes 4 bracelets using an average of 3 beads each. In the second session, the student completes 2 bracelets using an average of 9 beads each. Which of the following is the overall average number of beads per bracelet across both sessions?
A15
B5
C6
D12
Show answer & explanation
Let's work it out
1.Find the total number of beads in each session by multiplying the number of bracelets by that session mean: 4 × 3 = 12 beads in session 1 and 2 × 9 = 18 beads in session 2.
2.The grand total is 12 + 18 = 30 beads across 4 + 2 = 6 bracelets, giving an overall average of 30 ÷ 6 = 5 beads per bracelet.
Answer: B) 5
Question 3 · Arithmetic & Number SenseEasy
Pattern sequencePattern sequence: 2, 4, 5, 9, 8, 14, 11, 19, 14, 24, …. What is the 18th term? 2→4→5→9→8→14→11→19→14→24→…What is the 18th term?
The sequence below alternates between two interlocking patterns. What is the 18th term?
A38
B35
C44
D53
Show answer & explanation
Let's work it out
1.Identify the two interlocking subsequences by separating odd-positioned and even-positioned terms.
2.Odd-positioned terms (positions 1, 3, 5, 7, 9, 11, 13, 15, 17, …): 2, 5, 8, 11, 14, 17, 20, 23, 26, … — an arithmetic sequence with first term 2 and common difference 3.
3.Even-positioned terms (positions 2, 4, 6, 8, 10, 12, 14, 16, 18, …): 4, 9, 14, 19, 24, 29, 34, 39, 44, … — an arithmetic sequence with first term 4 and common difference 5.
4.Since 18 is even, the 18th term belongs to the even-positioned subsequence.
5.It is the 9th term of that subsequence.
6.Using the formula for the nth term of an arithmetic sequence: b_n = b_1 + (n − 1)d b_9 = 4 + (9 − 1)(5) = 4 + 40 = 44 The 18th term is 44.
Answer: C) 44
Question 4 · MathEasy
Material Thickness (meters)
ExpressionValue
5225
515
501
5-115\frac{1}{5}
5-2?
A carpenter records the thickness of various materials as powers of 5 meters. The table below shows several entries. What is the missing value in the table?
A-125\frac{1}{25}
B-25
C25
D125\frac{1}{25}
Show answer & explanation
Let's work it out
A negative exponent indicates a reciprocal: 5-2 = 1 divided by 52 = 1 divided by 25, which equals 125\frac{1}{25}.
Answer: D) 125\frac{1}{25}
Question 5 · Coordinate GraphsEasy
Cookie Baking Cost
Batches (x)Total Cost in dollars (y)
1350
2500
3650
4800
The table shows the total ingredient cost (in dollars) for baking different numbers of batches of cookies for a school fundraiser. Which equation models the total cost y as a linear function of the number of batches x?
Ay = 150x + 250
By = 150x + 350
Cy = 200x + 150
Dy = 150x + 200
Show answer & explanation
Let's work it out
1.The slope is found by computing the change in cost per additional batch: 500−3502−1\frac{500 - 350}{2 - 1} = 150, confirmed consistently across all rows.
2.Using the point (1, 350) in y = mx + b gives 350 = 150(1) + b, so b = 200.
3.The equation is y = 150x + 200, which correctly predicts all four values in the table.
Answer: D) y = 150x + 200
Question 6 · Data & StatisticsEasy
School Supply Shop Sales (in dollars)
MonthOnlineIn-Store
September1,200800
October1,5001,000
November2,100900
December1,8001,500
The table shows monthly sales figures (in dollars) for a school supply shop, broken down by online and in-store purchases over four months. What fraction of the shop's total four-month sales came from online purchases?
A29\frac{2}{9}
B718\frac{7}{18}
C1625\frac{16}{25}
D1118\frac{11}{18}
Show answer & explanation
Let's work it out
1.Add the online sales across all four months: 1,200 + 1,500 + 2,100 + 1,800 = 6,600 dollars.
2.Then add every month's total to get the grand total: (1,200 + 800) + (1,500 + 1,000) + (2,100 + 900) + (1,800 + 1,500) = 2,000 + 2,500 + 3,000 + 3,300 = 10,800 dollars.
3.The fraction of total sales that came from online purchases is 6,60010\frac{600}{10},800, which simplifies by dividing both numerator and denominator by 600 to give 1118\frac{11}{18}.
Answer: D) 1118\frac{11}{18}
Question 7 · Arithmetic & Number SenseStandard
The school auditorium has 12 rows of seats. The first row has 7 seats, and each row after the first has 4 more seats than the row in front of it. What is the total number of seats in the auditorium?
A51
B297
C372
D348
Show answer & explanation
Let's work it out
1.The row counts form an arithmetic sequence with first term 7 and common difference 4.
2.The 12th row has 7 + 11 times 4 = 7 + 44 = 51 seats.
3.The total number of seats is the sum of the first 12 terms: S = 12 divided by 2, times (7 + 51) = 6 times 58 = 348.
Answer: D) 348
Question 8 · MathStandard
Evaluate [(1,000,000)-2 × (100,000)3] ÷ [(10,000)-1 × 1,000,000,000].
A10-2
B1022
C10-10
D1014
Show answer & explanation
Let's work it out
1.Rewrite the large numbers as powers of 10.
2.The numerator becomes 10-12 × 1015 = 103, while the denominator becomes 10-4 × 109 = 105.
3.Dividing gives 103 - 5 = 10-2.
Answer: A) 10-2
Question 9 · Fractions & DecimalsStandard
Which value is equal to 134+12+13\displaystyle \frac{1}{\frac{3}{4} + \frac{1}{2 + \frac{1}{3}}}?
A3328\frac{33}{28}
B113\frac{11}{3}
C2833\frac{28}{33}
D1219\frac{12}{19}
Show answer & explanation
Let's work it out
1.Evaluate from the innermost fraction outward: 2 + 13\frac{1}{3} = 73\frac{7}{3}, so 12+13\displaystyle \frac{1}{2 + \frac{1}{3}} = 37\frac{3}{7}.
2.Then 34\frac{3}{4} + 37\frac{3}{7} = 3328\frac{33}{28}, and taking the outer reciprocal gives 2833\frac{28}{33}.
Answer: C) 2833\frac{28}{33}
Question 10 · GeometryStandard
For a school festival, a cooking club prepares 180,000 cubic centimeters of cake batter and sets aside 45,000 cubic centimeters for cupcakes. The rest exactly fills a giant cake mold consisting of a cylindrical lower section and a hemispherical dome. Both sections have radius 30 centimeters. Using 3 for pi, what is the height, in centimeters, of the cylindrical section?
A46 23\frac{2}{3}
B30
C10
D50
Show answer & explanation
Let's work it out
1.After the cupcake batter is removed, the mold receives 135,000 cubic centimeters of batter.
2.The hemispherical dome holds 54,000 cubic centimeters, leaving 81,000 cubic centimeters for the cylinder.
3.Since the cylinder has base area 2,700 square centimeters, its height is 81,000 divided by 2,700, or 30 centimeters.
Answer: B) 30
Question 11 · ProbabilityStandard
At a school music festival, a listening challenge uses 1,000 prompt cards, 500 about melody and 500 about rhythm. After a melody card is used in round 1, the judge returns it to the card box because melody prompts may be repeated. After a rhythm card is used, the player keeps it as a point token. The cards are mixed before a card is chosen for round 2. What is the probability that the round 2 card is about melody?
A9992000\frac{999}{2000}
B19973996\frac{1997}{3996}
C500999\frac{500}{999}
D19993996\frac{1999}{3996}
Show answer & explanation
Let's work it out
1.Separate the outcome into two cases based on the first card.
2.The melody-then-melody case has probability 5001000\frac{500}{1000} times 5001000\frac{500}{1000}, while the rhythm-then-melody case has probability 5001000\frac{500}{1000} times 500999\frac{500}{999}.
3.Adding these probabilities gives 14\frac{1}{4} + 250999\frac{250}{999} = 19993996\frac{1999}{3996}.
Answer: D) 19993996\frac{1999}{3996}
Question 12 · Ratios & ProportionsStandard
At tennis practice, two automatic launchers are used for a return drill. Launcher A sends a ball across the court at a steady rate of 1 ball every 4 seconds. It runs alone for 20 seconds before launcher B begins sending 1 ball every 6 seconds. Both then run until they have launched 30 balls altogether. How many more balls does launcher A launch than launcher B?
A10
B5
C9
D0
Show answer & explanation
Let's work it out
1.Launcher A sends 5 balls before launcher B starts, leaving 25 balls to be launched.
2.Their combined rate is 512\frac{5}{12} ball per second, so launching the remaining balls takes 60 seconds.
3.In that time A sends 15 balls and B sends 10, making their final totals 20 and 10; the difference is 10.
Answer: A) 10
Question 13 · Fractions & DecimalsStandard
While tuning a digital track, an editor tests negative values of n for a pitch adjustment. Which value of n makes 34−1n12+1n\displaystyle \frac{\frac{3}{4} - \frac{1}{n}}{\frac{1}{2} + \frac{1}{n}} equal to 4?
A-85\frac{8}{5}
B-125\frac{12}{5}
C-207\frac{20}{7}
D-4
Show answer & explanation
Let's work it out
1.Set x = 1n\frac{1}{n} so the equation becomes 34−x12+x\displaystyle \frac{\frac{3}{4} − x}{\frac{1}{2} + x} = 4.
2.Cross-multiply: 34\frac{3}{4} − x = 4(12\frac{1}{2} + x) = 2 + 4x.
3.Subtract 4x from both sides and subtract 34\frac{3}{4} from both sides: −5x = 2 − 34\frac{3}{4} = 54\frac{5}{4}, so x = −14\frac{1}{4}.
4.Since x = 1n\frac{1}{n}, we have 1n\frac{1}{n} = −14\frac{1}{4}, which gives n = −4.
Answer: D) -4
Question 14 · GeometryStandard
In a puzzle game, a token advances only if the three selected lengths form a right triangle whose hypotenuse is also a leg of a second right triangle with other leg 12.0 and hypotenuse 15.0. Which set of lengths advances the token?
A7.2, 9.6, 12.0
B1.8, 2.4, 3.0
C48.6, 64.8, 81.0
D5.4, 7.2, 9.0
Show answer & explanation
Let's work it out
1.The shared side satisfies h² + 12.0² = 15.0², so h² = 81 and h = 9.0.
2.Since 5.4² + 7.2² = 29.16 + 51.84 = 81, the required set is 5.4, 7.2, 9.0.
Answer: D) 5.4, 7.2, 9.0
Question 15 · Equations & InequalitiesHard
For a cooking contest, each batch makes 2 trays of 4 tasting bites, and 5 bites are set aside for the judges. The team must have fewer than 51 bites left for visitors. The team begins with 25 cups of dough, uses 3 cups per batch, and must have fewer than 10 cups left. Which candidate number of batches meets both requirements?
A6
B4
C7
D8
Show answer & explanation
Let's work it out
1.The number of visitor bites is 8x - 5, so 8x - 5 < 51 gives x < 7.
2.The remaining dough is 25 - 3x, so 25 - 3x < 10 gives x > 5 after the inequality reverses.
3.Thus 5 < x < 7, and the candidate that meets both requirements is 6.
Answer: A) 6
Question 16 · Algebraic ExpressionsHard
If P(x) + (12\frac{1}{2}x - 23\frac{2}{3})(34\frac{3}{4}x + 56\frac{5}{6}) = 78\frac{7}{8}x2 - 112\frac{1}{12}x + 19\frac{1}{9}, what is P(x)?
A12\frac{1}{2}x2 + 23\frac{2}{3}
B54\frac{5}{4}x2 - 16\frac{1}{6}x - 49\frac{4}{9}
C12\frac{1}{2}x2 - 112\frac{1}{12}x + 23\frac{2}{3}
D12\frac{1}{2}x2 - 49\frac{4}{9}
Show answer & explanation
Let's work it out
1.Expanding the product gives 38\frac{3}{8}x2 - 112\frac{1}{12}x - 59\frac{5}{9}.
2.Subtract this polynomial from 78\frac{7}{8}x2 - 112\frac{1}{12}x + 19\frac{1}{9} and combine corresponding terms.
Answer: A) 12\frac{1}{2}x2 + 23\frac{2}{3}
Why the other choices are tempting: The result is P(x) = 12\frac{1}{2}x2 + 23\frac{2}{3}.
Question 17 · MathHard
During a classroom activity, students compare the exponential function E(x) = (12\frac{1}{2})rx with the linear function L(x) = 12\frac{1}{2} + 5x/16. Which candidate value of r makes both E(2) = L(2) and E(3) > L(3) true?
A516\frac{5}{16}
B94\frac{9}{4}
C32\frac{3}{2}
D98\frac{9}{8}
Show answer & explanation
Let's work it out
1.Evaluating the linear function gives L(2) = 98\frac{9}{8}, so (12\frac{1}{2})r2 = 98\frac{9}{8} and r2 = 94\frac{9}{4}.
2.Testing the resulting candidates at x = 3 shows that r = 32\frac{3}{2} gives E(3) = 2716\frac{27}{16}, which is greater than L(3) = 2316\frac{23}{16}.
Answer: C) 32\frac{3}{2}
Question 18 · Equations & InequalitiesHard
What is the sum of the squares of all real numbers x that satisfy |2|x - 3| - 5| = 1?
A62
B61
C26
D36
Show answer & explanation
Let's work it out
1.The outer absolute value produces the cases 2|x - 3| - 5 = 1 and 2|x - 3| - 5 = -1.
2.These simplify to |x - 3| = 3 and |x - 3| = 2, giving x = 6, 0, 5, and 1.
3.The sum of their squares is 36 + 0 + 25 + 1 = 62.
Answer: A) 62
Question 19 · Algebraic ExpressionsHard
In a road-trip game, the lengths of two route legs are x+m miles and x+n miles, and their product is the route score. Which candidate score can be factored in this form if m and n are distinct positive whole numbers and m+n is one-half of mn?
Ax2 + 5x + 4
Bx2 + 9x + 18
Cx2 + 8x + 16
Dx2 + 4x + 4
Show answer & explanation
Let's work it out
1.Rewrite the condition as mn-2m-2n=0, then add 4 to both sides to obtain (m-2)(n-2)=4.
2.The positive factor pair that produces distinct values gives m=3 and n=6.
3.Their product is (x+3)(x+6)=x2+9x+18.
Answer: B) x2 + 9x + 18
Question 20 · MathHard
For h(x)=x+7\sqrt{x+7}+−x−2\sqrt{-x-2}+1x+4\frac{1}{x+4}, what is the sum of all integer inputs in the domain of h?
A-23
B-27
C-2
D-14
Show answer & explanation
Let's work it out
1.The square roots require x ≥ -7 and x ≤ -2, so their combined restriction is -7 ≤ x ≤ -2.
2.Since the denominator cannot be zero, -4 must be removed.
3.The remaining integer inputs are -7, -6, -5, -3, and -2, whose sum is -23.
Answer: A) -23

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